which amounts to setting \(\mathbf {f}({\mathbf {A}})={\mathbf {A}}^{-1}\)  [1]}. The Krylov subspace is characterized by its dimension \(\ell \) , matrix \({\mathbf {A}}\) , and vector \(\mathbf {b}\)  and is denoted by \(\mathcal {K}_\ell ({\mathbf {A}},\mathbf {b})\) . Its construction proceeds as follows: vector \(\mathbf {b}\)  is left multiplied by \({\mathbf {A}}\) , which results in a new vector \({\mathbf {A}}\mathbf {b}\) . The new vector is multiplied again with matrix \({\mathbf {A}}\)  to find \({\mathbf {A}}^2\mathbf {b}\) , and this goes on \(\ell -1\)  times, vector \(\mathbf {b}\)  being included in the subspace:
\(\mathcal {K}_\ell ({\mathbf {A}},\mathbf {b})=\mathrm {span} \big \lbrace \mathbf {b},{\mathbf {A}}\mathbf {b},{\mathbf {A}}^2\mathbf {b},...,{\mathbf {A}}^{\ell -1}\mathbf {b}\big \rbrace .\) 
